Limit at Zero of the Brownian First-Passage Density
نویسنده
چکیده
(1.1) = inf f t > 0 j Bt g(t) g is exponentially distributed (i.e. has a density function f(t) = e t with t > 0 and > 0 ). This problem can be viewed as an inverse to the problem of finding a density function f of when g is given. Explicit solutions to the latter problem are known only in a limited number of special cases including linear or quadratic g . The law of is also known for a square-root boundary g but only in the form of a Laplace transform (which appears intractable to inversion). The inverse problem seems even harder. While it is relatively easy to use a comparison argument and rule out those g satisfying g(0+) > 0 (since in this case f(0+) = 0 ), it is much less obvious to see if there is a continuous function g at all for which f(0+) is strictly positive and finite. The knowledge of f(0+) in terms of g , on the other hand, is of interest in various numerical methods found in the literature for computing f when g is known.
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